Annual Report 2000 - WIT

a
141
only. The quickest possible configuration of the phase front is usually given by the
Hamilton-Jacobi, i.e., eikonal equation. However, the conditions above necessarily
take into account not only kinematic aspects of the front propagation but rather mainly
dynamic aspects, i.e., amplitude of the pressure perturbation. In other words, the
eikonal equation is usually valid in much broader domain of frequencies than those
given by the inequalities above. Therefore, the method will give meaningful and
useful results, at least semi qualitatively.
To verify the assertion that we are able to describe the diffusion process and the
kinematics of the evolution of the triggering front in a heterogeneous medium by
the use of the eikonal solution, we performed some numerical tests. We solved the
parabolic differential equation of diffusion (1) in two spatial dimensions with the help
of a finite elements method (FE) implemented in the MATLAB R
computing environment.
We calculated the time-dependent pressure variation through a medium where
the » diffusivity varies smoothly Ê in direction with a Gaussian profile. It changes
from a background » ‘uÎÄÈ;ÍW¢ËÒ? value of to a minimal value »ú‘Î–È Ìè¢ËÒÆ of at the
¨
¨
center. The half-width of this heterogeneity is approx. 200 m. The dimension of the
computational mesh is 4000 m x 4000 m, and the source point is located at its center.
As input signal we use a time-harmonic sinusoidal signal with a period of 400 h and
800 h respectively, multiplied with a boxcar function for the whole simulation time of
2400 h. The time increment in our ‡Pˆb_ Þ simulations was h while the elementary
cell was of the order of 5 m x 5 m. We observe the pressure variation in a one dimensional
section along the x-axis through the center of the model, where the source is
located. In each point of observation we estimate the arrival time of the fourth resp.
sixth zero-crossing of our quasi-periodic pressure signal. This was necessary to reduce
the effects of the high-frequency components due to the finite character of the source
signal. We compare this time with the corresponding theoretical eikonal solution
‘Sc¢d
ä
ˆ
R%ë
Œ°šŠëŸ
È (32)
Using the arrival time we also calculate the velocity Πof the phase front at a given
distance from the source point. Then we convert it into the diffusivity and compare
the result with the exact diffusivity of the model. The diffusivity is calculated from the
measured velocity of the phase front by
»ÃšŠÊƒŸ ‘
ŒƒšŠÊƒŸFÒ
£=e §
• (33)
e where is the dominant frequency of the source function (see eq. 4 and the
comment below).